An Ultrafinite Set Theory
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From: William Elliot on 2 Mar 2010 02:33 On Mon, 1 Mar 2010, RussellE wrote: > On Mar 1, 12:14�am, William Elliot <ma...(a)rdrop.remove.com> wrote: >>>>> 7) Axiom of finiteness: There is a largest and smallest urelement. >> >>>> That doesn't make U finite. The ordinal number omega_0 + 1 >>>> has a smallest and largest element and isn't finite. >> >>> Yes, I know. I am still having problems coming up with >>> an axiom of finiteness. >> >> You could include in the language, k constant symbols u1,.. uk, >> define U = { u1,.. uk } and state that if u is an urelement, >> then u in U. > > This seems to be the simplest solution. > It would be nice to have something more "elegant". > Simple is elegant. >>>>> The singleton axiom and union axiom are enough to create any set. >>>> No. Even assuming U is finite, you can't construct an empty set. >>> I think I can derive that from intersection. >> >> You can't if there's only one urelement. > > Yes. I am not sure this is a problem. > I could add an empty set axiom. > I want to minimize the number of axioms. > > If I remember correctly, if an axiomatic set theory is > consistent, it is still consistent when we negate an axiom. It is not. You can delete an axiom but to negate an axiom you first have to prove that each axiom is independent of the others. > I am not sure how I can deal with an anti-empty set axiom. > Don't. If you don't need an empty set, then don't create one. > I use to think anti-foundational set theories were strange. > Lately, I have been considering anti-union theories and > anti-comprehension theories. There exists two sets with > the same elements that are not equal. > Read about fuzzy set theory. >> It also excludes the positive integers of Piano's axiom. > > Of course. It's not an UST if it doesn't exclude these. > Then don't call them natural numbers. That expression has been taken. Call them something else. >> Your natural numbers are unnatural. �If it doesn't smell >> like a dog nor bark or look like a dog, then it isn't a dog. > > Are you saying my UST is "counter-intuitive"? No. I'm saying don't call it the natural numbers as they aren't natural and natural numbers already means something that can't be your numbers. Call them something else, like natural computer numbers. > I find it amusing that a finite set theory is "counter-intuitive". > What's finite set theory? The theory of finite sets? > We all have pre-conceived intuitions about "natual numbers". > I don't think natural numbers can grow without limit. That's because you're limited by the lack of visualization of computers. > I want my set theory to formalized my pre-conceived notion > of natural numbers. I myself, have consider bounded integers and found the complexity too much to be of worth. >>>>> Many people have told me all known UST's are inconsistent. >>>>> Obviously, no UST will be consistent with axioms from other >>>>> set theories. No UST will be consistent with the axiom >>>>> "if n is a natural number then n+1 is a natural number". >>>>> My UST doesn't have this axiom. >> >>>> Of course it doesn't. You haven't even defined incrementation. >> >>> I have a well ordering axiom. What else do I need? >> >> A definition of n + 1 as the successor urelement. > > How does ZFC define the successor function? S(x) = x \/ {x} > Is there a "successor" axiom? It's one of Peano's axioms. > "n+1" is meaningless for certain n in my UST. > Which ones? You could call some large urelement oo (ie infinity) or overflow and instead of a + b and S(u) being undefined for certain urelements, you could say a + b = oo and S(oo) = oo. > Some people think the universe is a computer. > If so, there are numbers too big and too small > for the universe to comprehend. > Their universe excludes the mind. > Position and momentum can't be computed beyond a certain precision. > Commonplace physics. The national debt can't be computer beyond a certain percision also. What happens when you've a finite set of integers and some physicists has need for a larger or more precise number that what you provide? > It could be worse. If physicists come up with a set theory it will be > something like "there is a probability 1=1, a probability 1=2, ..." > Look into fuzzy set theory. ----
From: RussellE on 2 Mar 2010 03:16 On Mar 1, 11:33 pm, William Elliot <ma...(a)rdrop.remove.com> wrote: > On Mon, 1 Mar 2010, RussellE wrote: > > On Mar 1, 12:14 am, William Elliot <ma...(a)rdrop.remove.com> wrote: > >>>>> 7) Axiom of finiteness: There is a largest and smallest urelement. > > >>>> That doesn't make U finite. The ordinal number omega_0 + 1 > >>>> has a smallest and largest element and isn't finite. > > >>> Yes, I know. I am still having problems coming up with > >>> an axiom of finiteness. > > >> You could include in the language, k constant symbols u1,.. uk, > >> define U = { u1,.. uk } and state that if u is an urelement, > >> then u in U. > > > This seems to be the simplest solution. > > It would be nice to have something more "elegant". > > Simple is elegant. > > >>>>> The singleton axiom and union axiom are enough to create any set. > >>>> No. Even assuming U is finite, you can't construct an empty set. > >>> I think I can derive that from intersection. > > >> You can't if there's only one urelement. > > > Yes. I am not sure this is a problem. > > I could add an empty set axiom. > > I want to minimize the number of axioms. > > > If I remember correctly, if an axiomatic set theory is > > consistent, it is still consistent when we negate an axiom. > > It is not. You can delete an axiom but to negate an axiom > you first have to prove that each axiom is independent of > the others. Thanks. > > I am not sure how I can deal with an anti-empty set axiom. > > Don't. If you don't need an empty set, then don't create one. > > > I use to think anti-foundational set theories were strange. > > Lately, I have been considering anti-union theories and > > anti-comprehension theories. There exists two sets with > > the same elements that are not equal. > > Read about fuzzy set theory. > > >> It also excludes the positive integers of Piano's axiom. > > > Of course. It's not an UST if it doesn't exclude these. > > Then don't call them natural numbers. > That expression has been taken. Call them something else. > Call > them something else, like natural computer numbers. OK. We can call them natural computer numbers. > > I want my set theory to formalized my pre-conceived notion > > of natural numbers. > > I myself, have consider bounded integers and > found the complexity too much to be of worth. Most computer engineers agree with you. > >>>>> Many people have told me all known UST's are inconsistent. > >>>>> Obviously, no UST will be consistent with axioms from other > >>>>> set theories. No UST will be consistent with the axiom > >>>>> "if n is a natural number then n+1 is a natural number". > >>>>> My UST doesn't have this axiom. > > >>>> Of course it doesn't. You haven't even defined incrementation. > > >>> I have a well ordering axiom. What else do I need? > > >> A definition of n + 1 as the successor urelement. > > > How does ZFC define the successor function? > > S(x) = x \/ {x} My theory won't allow the union of an element and a set. I can define successor as a "circuit". Assume we have the set U = {a,b,c,d}. We arbitrarily choose a singleton set like {a}. Define the variable K_in to be true if a set has k as an element. Define K_out to be true if the successor has k as an element. A_out = B_in B_out = C_in C_out = D_in D_out = A_in This is a successor function for the elements of U. It doesn't actually define the "first" element. > > Is there a "successor" axiom? > > It's one of Peano's axioms. > > > "n+1" is meaningless for certain n in my UST. > > Which ones? You could call some large urelement oo (ie infinity) > or overflow Programmers use NaN. Not a number. > and instead of a + b and S(u) being undefined for certain > urelements, you could say a + b = oo and S(oo) = oo. The simplest is to define "0" as a successor. I can also define modulo arithmetic. > > Some people think the universe is a computer. > > If so, there are numbers too big and too small > > for the universe to comprehend. > > Their universe excludes the mind. > > > Position and momentum can't be computed beyond a certain precision. > > Commonplace physics. The national debt can't be computer beyond a > certain percision also. What happens when you've a finite set > of integers and some physicists has need for a larger or more > precise number that what you provide? Add another urelement. > > It could be worse. If physicists come up with a set theory it will be > > something like "there is a probability 1=1, a probability 1=2, ..." > > Look into fuzzy set theory. I never found much use for fuzzy logic. It works well for some things, but, knowing something is 80% true doesn't help in a lot of situations. There are easier ways to calculate odds than fuzzy logic. I like multi-valued logics. A tri-value logic with "true", "false", and "don't know" is interesting. Russell - 2 many 2 count
From: William Elliot on 2 Mar 2010 06:37 On Tue, 2 Mar 2010, RussellE wrote: >>> On Mar 1, 12:14�am, William Elliot <ma...(a)rdrop.remove.com> wrote: >>>> It also excludes the positive integers of Piano's axiom. >>> Of course. It's not an UST if it doesn't exclude these. >> >> Then don't call them natural numbers. >> That expression has been taken. �Call them something else. >> Call them something else, like natural computer numbers. > > OK. We can call them natural computer numbers. > >>> I want my set theory to formalized my pre-conceived notion >>> of natural numbers. >> >> I myself, have consider bounded integers and >> found the complexity too much to be of worth. > > Most computer engineers agree with you. > > I can define successor as a "circuit". What's a circuit? > Assume we have the set U = {a,b,c,d}. > We arbitrarily choose a singleton set like {a}. > Define the variable K_in to be true if a set has k as an element. Crazy. What you doing? Your syntax doesn't make any sense. For each urelement you're labeling a variable? No. Doesn't even make basic programming sense. You are defining a proposition over urelements. I(k) when k in U and some set A with k in A. > Define K_out to be true if the successor has k as an element. > That defies any sense whatever. Are you sure you wrote what you intended to write? > A_out = B_in > B_out = C_in > C_out = D_in > D_out = A_in > Meaningless other than it seems to be a circuit, a cycle. Sure you could give U a circular order like the integers modulus |U| > 1. > This is a successor function for the elements of U. > It doesn't actually define the "first" element. Since U is well ordered, isn't the successor of k the successor of k in the well ordering? >>> Is there a "successor" axiom? >> It's one of Peano's axioms. >> >>> "n+1" is meaningless for certain n in my UST. >> >> Which ones? You could call some large urelement oo (ie infinity) >> or overflow > > Programmers use NaN. Not a number. > >> and instead of a + b and S(u) being undefined for certain >> urelements, you could say a + b = oo and S(oo) = oo. > > The simplest is to define "0" as a successor. > I can also define modulo arithmetic. > Make up your mind. >>> Position and momentum can't be computed beyond a certain precision. >> >> Commonplace physics. The national debt can't be computer beyond a >> certain precision also. What happens when you've a finite set >> of integers and some physicists has need for a larger or more >> precise number that what you provide? > > Add another urelement. > How would the others using your hippy numbers know that it was done and what it was. How would you convert all previous work to the new augmented yippy numbers. Most of all, who decides when and if more urelements need to be included, how many are needed and what new urelement will be? I can see it now: Natural Computer Numbers 2011.3.10. >>> It could be worse. If physicists come up with a set theory it will be >>> something like "there is a probability 1=1, a probability 1=2, ..." >> >> Look into fuzzy set theory. > > I never found much use for fuzzy logic. > It works well for some things, but, knowing something > is 80% true doesn't help in a lot of situations. If you knew that the US dollar was about to collapsed with an 80% assurance you'd keep 20% of your money in US dollars and 80% in gold or in food, land and guns, depending upon how much of an optimist you are. > There are easier ways to calculate odds than fuzzy logic. > > I like multi-valued logics. A tri-value logic with > "true", "false", and "don't know" is interesting. > You'd also like the tribe that counts one, two, many. Most language count one, many; singular and plural. Sanskrit counts, one, two, many; singular, dual and plural. s + s = d; s+d = d+d = s+p = d+p = p+p = p S(s) = d; S(d) = p; S(p) = p
From: Jesse F. Hughes on 2 Mar 2010 09:38 RussellE <reasterly(a)gmail.com> writes: > On Mar 1, 1:48 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: >> On Feb 28, 6:34 pm, RussellE <reaste...(a)gmail.com> wrote: >> >> > Simpler is better. Here is a simple ultrafinite set theory (UST). >> >> > Primitives: >> >> > Urelement - an element of a set. A set or proper class can not be an >> > urlelement. >> > Set - a collection of urelements. >> > Proper Class - a collection of sets. >> >> If they're primitives, then what is the part following the dash >> symbol? > >> Are those definitions or axioms or combination above? Are the >> primitives 'collection' and 'element'? Or what? > > OK. The primitives are element and collection. It seems to me that you also have predicates for Urelement, Set and Class. > > urelement - Only objects defined to be urelements can be elements of a > set > set - A collection of elements. > proper class - a collection of sets. So, it seems to me that you want the following axioms (Ax)(Urelement(x) -> (Ay)~y e x) (Ax)(Set(x) -> (Ay)(y e x -> Urelement(y)) (Ax)(Class(x) -> (Ay)(y e x -> Set(y)) (Ax)(Set(x) -> (~Urelement(x) & ~Class(x)) (Ax)(Class(x) -> ~Urelement(x)) >> > 6) Axiom of well ordering: The urelements are well ordered. >> >> Assuming the ordinary definiton of 'well ordered', I guess. > > You got me. I don't define well ordering. > I can't define well ordering the way ZFC does. > My theory doesn't have sets of ordered pairs. > I could define proper classes as ordered pairs. > Any suggestions for a well ordering axiom would be welcome. Just add another relation < and axioms (Ax)(Ay)(x < y -> (Urelement(x) & Urelement(y))) and the usual axioms specifying that < is a well-ordering. >> > 7) Axiom of finiteness: There is a largest and smallest urelement. Simply add another axiom that <^op (i.e., >) is also a well-ordering. Every well-ordering has a least element. You'll have to check, of course, that you *can* write down the axioms for well-ordering. I don't see any issues, but I haven't thought it through. -- "At some point in the future history of humanity, AP will eclipse even Jesus." -- Archimedes Plutonium, 10/21/07 "I wrote those lines because I am not a megalomania [sic] but rather very humble and down to earth." -- Archimedes Plutonium, 10/22/07
From: MoeBlee on 2 Mar 2010 13:03
On Mar 2, 12:48 am, RussellE <reaste...(a)gmail.com> wrote: > On Mar 1, 1:48 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > OK. The primitives are element and collection. > > urelement - Only objects defined to be urelements can be elements of a > set > set - A collection of elements. > proper class - a collection of sets. That is quite confused. The following is the best I can make sense of what you might be driving at. (By the way, the reason yours is confused is not because of the words 'set', etc., that you use but rather your table of the words is mixed up. However, I've proposed new words since there is no sense in confusing with the ordinary use of words in set theory): Primitives: 1-place predicate - 'x is a ret' 1-place predicate - 'x is an urment' 2-place predicate - 'xey' ('x is an element of y') Axiom: x is a ret -> Ay(yex -> y is an urment) Definition: x is a rass <-> Ay(yex -> y is a ret) (So you don't need 'collection'.) Maybe you still want to revise what I did, but at least mine is clear. > > > 6) Axiom of well ordering: The urelements are well ordered. > > > Assuming the ordinary definiton of 'well ordered', I guess. > > You got me. I don't define well ordering. > I can't define well ordering the way ZFC does. Sure you can. Definitions of these kinds of predicates (in a language with 'e') don't depend on axioms. > 7) Axiom of finitenes: The set U = {u_0, u_1, ..., u_k} exists. Nope. What are '0', '1'? What does "..." mean? You're basically question begging by using "..." in this way to mean something like "finite" when what you need to do is DEFINE 'finite' (I'd call it 'r-nite') in your language. > > What is the purpose of your theory? > > I want to show it is possible to have a consistent, finite set theory. What do you MEAN by a 'finite theory'? You seem to have your own definition of 'finite'. (1) Do you mean (given the ordinary definition of 'finite'), a theory that has a theorem that there exist only finite sets? Then that is easy: The theory, in the language of set theory, whose sole axiom is "There exist only finite sets" is a consistent theory. (2) Do you mean (given the ordinary definition of 'finite'), a theory that has only models with finite domains? Again, that is easy. The theory, in the langauge of set theory, whose sole axiom is "Axy(x=y)" is consistent and has models only with finite domains. So what? > > Do you think it makes ordinary set > > theory otiose? > > No. Why would you think that? I'm just asking? > What do you mean by "ordinary mathematics for the sciences"? I have no PRECISE definition. It is left open to reasonable interpretation. But a typical minimum would be some calculus for predicting the motions of objects, for caclulating probabilites and for statistics. > Can you derive E=MC^2 from ZFC? As I understand, that is a statement of physics, not just of mathematics. I'm referring to the mathematics uses for physics, not the physics itself. > If you mean Peano arithematic, No, I don't. First order PA by itself, is, as far as I know, not adequate for a theory for the sciences. > > > I don't need an axiom schema of specification. > > > The singleton axiom and union axiom are enough to create any set. > > > Depends on what you think are "enough" sets. > > Do we really need a continuum number of sets to do "science"? I don't know. But that doesn't obviate the sense of my question as to what you consider enough sets. Why do you need sets at all? Why do you need a set theory? Why do you need any theory at all? What is your purpose in having a theory (other than merely to have a consistent one such that (1) or (2) from above) or in having a foundational theory? MoeBlee |